First, we examine (see e.g. [1]})
\(\begin{split}{\Big |{\frac{\mathrm {d}^{l}}{\mathrm {d}t^{l}}}\widetilde{\varphi }_j(tzV_n^{-1})\Big |}\quad \\&=\Big |\widetilde{\mathbb {E}}\Big [\big (izV_n^{-1}X_j\big )^l\exp (itzV_n^{-1}X_j)\Big ]\Big |\\&\le \widetilde{\mathbb {E}}\Big [z^lV_n^{-l}|X_j|^l\Big ]= z^lV_n^{-l}|X_j|^l\end{split}\) 
